A213640 E.g.f. satisfies: A(x) = x - log(1 - A(x)^2).

1, 2, 12, 132, 2040, 40560, 986160, 28344960, 940222080, 35350378560, 1485586206720, 69006955691520, 3510875174526720, 194162144086310400, 11597083480958976000, 744005639375065267200, 51024015181398702643200, 3725042532308649628876800, 288434288836744276094668800

Offset: 1

Author

Paul D. Hanna, Jun 17 2012

Keywords

• nonn

Examples

			E.g.f: A(x) = x + 2*x^2/2! + 12*x^3/3! + 132*x^4/4! + 2040*x^5/5! +...
Related series:
A(x)^2 = 2*x^2/2! + 12*x^3/3! + 120*x^4/4! + 1800*x^5/5! + 35280*x^6/6! +...
-log(1-A(x)^2) = 2*x^2/2! + 12*x^3/3! + 132*x^4/4! + 2040*x^5/5! +...
		

Crossrefs

Cf. A213641, A218652, A200319.

Programs

Mathematica

Rest[CoefficientList[InverseSeries[Series[x + Log[1-x^2],{x,0,20}],x],x] * Range[0,20]!] (* Vaclav Kotesovec, Jan 10 2014 *)

PARI

{a(n)=n!*polcoeff(serreverse(x+log(1-x^2 +x*O(x^n))), n)}
for(n=1,25,print1(a(n),", "))

PARI

{Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=x); A=x+sum(m=1, n, Dx(m-1, (-log(1 - x^2+x*O(x^n)))^m)/m!); n!*polcoeff(A, n)}

PARI

{Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=x+x^2+x*O(x^n)); A=x*exp(sum(m=1, n, Dx(m-1, (-log(1 - x^2+x*O(x^n)))^m/x)/m!)+x*O(x^n)); n!*polcoeff(A, n)}

Formula

E.g.f. A(x) satisfies:

(1) A(x + log(1-x^2)) = x.

(2) A(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) (-log(1-x^2))^n / n!.

(3) A(x) = x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) (-log(1-x^2))^n/x / n! ).

a(n) = n*A213641(n-1).

a(n) ~ sqrt(1-1/sqrt(2)) * n^(n-1) / (exp(n) * (sqrt(2) -1 + log(2*(sqrt(2)-1)))^(n-1/2)). - Vaclav Kotesovec, Jan 10 2014